1. Let U24 = {z E C|224 = 1}. Define £24= cos () + i sin (5).(a) List all subgroups of U24 and draw its lattice diagram.(b) Let W = ((£24)20) < U24. What is [U24 : W]?(c) Enumerate all left cosets of W in U24.

The given question is related with group theory. Given that U24 = z∈ℂ | z24 = 1 . Define ξ24 =…

Answered: 1. Let U24 = {z € C|224 = 1}. Define…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter2: The Integers

Section2.3: Divisibility

Problem 30E: Let be as described in the proof of Theorem. Give a specific example of a positive element of .

See similar textbooks

Similar questions

In Exercises 3 and 4, let G be the octic group D4=e,,2,3,,,, in Example 12 of section 4.1, with its multiplication table requested in Exercise 20 of the same section. Let H be the subgroup e, of the octic group D4. Find the distinct left cosets of H in D4, write out their elements, partition D4 into left cosets of H, and give [D4:H]. Find the distinct right cosets of H in D4, write out their elements, and partition D4 into right cosets of H. Example 12 Using the notational convention described in the preceding paragraph, we shall write out the dihedral group D4 of rigid motions of a square The elements of the group D4 are as follows: 1. the identity mapping e=(1) 2. the counterclockwise rotation =(1,2,3,4) through 900 about the center O 3. the counterclockwise rotation 2=(1,3)(2,4) through 1800 about the center O 4. the counterclockwise rotation 3=(1,4,3,2) through 2700 about the center O 5. the reflection =(1,4)(2,3) about the horizontal line h 6. the reflection =(2,4) about the diagonal d1 7. the reflection =(1,2)(3,4) about the vertical line v 8. the reflection =(1,3) about the diagonal d2. The dihedral group D4=e,,2,3,,,, of rigid motions of the square is also known as the octic group. The multiplication table for D4 is requested in Exercise 20 of this section.
In Exercises 3 and 4, let be the octic group in Example 12 of section 4.1, with its multiplication table requested in Exercise 20 of the same section. Let be the subgroup of the octic group . Find the distinct left cosets of in , write out their elements, partition into left cosets of , and give . Find the distinct right cosets of in , write out their elements, and partition into right cosets of . Example 12 Using the notational convention described in the preceding paragraph, we shall write out the dihedral group of rigid motions of a square The elements of the group are as follows: 1. the identity mapping 2. the counterclockwise rotation through about the center 3. the counterclockwise rotation through about the center 4. the counterclockwise rotation through about the center 5. the reflection about the horizontal line 6. the reflection about the diagonal 7. the reflection about the vertical line 8. the reflection about the diagonal . The dihedral group of rigid motions of the square is also known as the octic group. The multiplication table for is requested in Exercise 20 of this section.
Let be as described in the proof of Theorem. Give a specific example of a positive element of .
Consider the elliptic-curve group defined by { (x,y) | x,y ∈ Z7 and x2 mod 7 = x3 + 2x +3 mod 7 } (ie, the group you get when a=2, b=3, and p=7). What is (2,1) + (3,1) in this group? Write your answer as an ordered pair of integers with no spaces.
Consider the elliptic-curve group defined by { (x,y) | x,y ∈ Z11 and x2 mod 11 = x3 + 2x + 5 mod 11 } (ie, the group you get when a=2, b=5, and p=11). What is (3,4) + (8,4) in this group? What value do you get for s?______ What value do you get for x3?______ What value do you get for y3?_______
Suppose φ: ℤ x ℤ→ℤ defined by φ(m, n) = m - n. a. Compute Ker φ and φ(Z). b. To what group is ℤ→ℤ / Ker φ isomorphic to?
Obtain al the Sylow p-subgroups of (Z/2Z) X S3
Consider the group G = ℚ* × ℤ with operation * on G that can be expressed as: (w, x) * (y, z) = (wy + 1, xz - 1), for all (w, x), (y, z) ∈ ℚ* × ℤ. Is the group <G, *> abelian?
In D12 = <x, y | x2 = y12 = e, xyx = y-1>, prove that the subgroupH = <x, y3> (which is isomorphic to D4) is not a normal subgroup.
find the galois group of x^4 + 2x^2 + x + 3 over Q.
Determine the order of (Z ⨁ Z)/<(2, 2)>. Is the group cyclic?
[Zp-(0),.] Where p is prime is an abelian group

SEE MORE QUESTIONS

Recommended textbooks for you

Elements Of Modern Algebra

Algebra

ISBN:

9781285463230

Author:

Gilbert, Linda, Jimmie

Publisher:

Cengage Learning,

Elements Of Modern Algebra

Algebra

ISBN:

9781285463230

Author:

Gilbert, Linda, Jimmie

Publisher:

Cengage Learning,

SEE MORE TEXTBOOKS

1. Let U24 = {z € C|224 = 1}. Define $24 = cos () + i sin (5). %3D (a) List all subgroups of U24 and draw its lattice diagram. (b) Let W = ((§24)20) < U24. What is [U24 : W]? (c) Enumerate all left cosets of W in U24.